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→Scales: stretch & compression
 
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[[File:blue_60edo.png|alt=blue_60edo.png|blue_60edo.png]]
[[File:blue_60edo.png|alt=blue_60edo.png|blue_60edo.png]]


== Scales ==
== Octave stretch or compression ==
* [[5- to 10-tone scales in 60edo]]
What follows is a comparison of compressed- and stretched-octave 60edo tunings.
* Amulet{{idiosyncratic}} (approximated from [[25edo]], subset of [[magic]]): 5 2 5 5 2 5 7 5 5 2 5 7 5
* Approximations of [[gamelan]] scales:
** 5-tone pelog: 6 8 20 5 21
** 7-tone pelog: 6 8 12 8 5 14 7
** 5-tone slendro: 12 12 12 12 12
 
== Nearby equal-step tunings ==
There are a few other useful [[equal-step tuning]]s which occur close to 60edo in step size:
 
; 207ed11, 168ed7
 
The tunings [[207ed11]] and [[168ed7]] are almost identical. Each is 60edo but with slightly ''stretched'' octaves.  


Each induces relatively large improvement to [[5/1]], [[7/1]] and [[11/1]] at the cost of worsening of [[2/1]] and [[3/1]]. Each also causes the [[patent val]] to flip for [[11/1]] and [[13/1]].
60edo can benefit from slightly [[stretched and compressed tuning|stretching the octave]], especially when using it as a no-11 17-limit equal temperament. With the right amount of stretch we can find better harmonics 3, 5, and 7 at the expense of somewhat less accurate approximations of 2 and 13. Tunings such as [[95edt]] or [[155ed6]] are great demonstrations of this.
{{Harmonics in equal|207|11|1|intervals=prime|columns=11|collapsed=1}}
{{Harmonics in equal|168|7|1|intervals=prime|columns=11|collapsed=1}}


; 139ed5
; [[zpi|303zpi]]
* Step size: 19.913{{c}}, octave size: 1194.78{{c}}
Compressing the octave of 60edo by around 5{{c}} results in improved primes 5, 7 and 13, but worse primes 2, 3 and 11. This approximates all harmonics up to 16 within 8.75{{c}}. The tuning 303zpi does this. So does [[equal tuning|223ed13]] whose octave is identical within 0.03{{c}}.
{{Harmonics in cet|19.913|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 303zpi}}
{{Harmonics in cet|19.913|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 303zpi (continued)}}


The tuning [[139ed5]] is 60edo but with slightly ''stretched'' octaves.  
; [[ed7|169ed7]]  
* Step size: 19.958{{c}}, octave size: 1197.50{{c}}
Compressing the octave of 60edo by around 2.5{{c}} results in improved primes 7 and 11, but worse primes 2, 3, 5 and 13. This approximates all harmonics up to 16 within 9.94{{c}}. The tuning 169ed7 does this.
{{Harmonics in equal|169|7|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 169ed7}}
{{Harmonics in equal|169|7|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 169ed7 (continued)}}


It induces relatively large improvement to 5/1, 7/1 and 11/1 at the cost of worsening of 2/1 and 13/1. It also causes the patent val for 11/1 to flip from 208 steps to 207 steps.
; [[zpi|302zpi]]
{{Harmonics in equal|139|5|1|intervals=prime|columns=11|collapsed=1}}
* Step size: 19.962{{c}}, octave size: 1197.72{{c}}
Compressing the octave of 60edo by around 2{{c}} results in improved primes 7 and 11, but worse primes 2, 3, 5 and 13. This approximates all harmonics up to 16 within 9.84{{c}}. The tuning 202zpi does this. So does the tuning [[equal tuning|208ed11]] whose octave is identical within 0.3{{c}}.
{{Harmonics in cet|19.962|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 302zpi}}
{{Harmonics in cet|19.962|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 302zpi (continued)}}


; 301zpi
302zpi is particularly well suited to [[catnip]] temperament specifically: in 60edo, catnip's mappings of 5 and 13 both differ from the [[patent val]]s, but in 19.95cet, only it's mapping of 7 differs. The tuning 169ed7 also does this, but 302zpi approximates most simple harmonics better than 169ed7.
 
The tuning [[301zpi]], the 301st [[zeta peak index]], is 60edo but with slightly ''stretched'' octaves.  
 
It induces relatively large improvement to 3/1, 5/1, 7/1, 11/1 and 17/1 at the cost of worsening of 2/1 and 13/1. It also causes the patent val for 11/1 to flip from 208 steps to 207 steps.
 
Like 60edo, 301zpi is distinctly consistent up to the [[integer limit|10-integer-limit]].
{{Harmonics in equal|1|38083|37645|intervals=prime|columns=11|title=Approximation of prime harmonics in 301zpi|collapsed=1}}


; 60edo
; 60edo
{{Harmonics in equal|60|2|1|intervals=prime|columns=11|collapsed=1}}
* Step size: 20.000{{c}}, octave size: 1200.00{{c}}
Pure-octaves 60edo approximates all harmonics up to 16 within 8.83{{c}}.
{{Harmonics in equal|60|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 60edo}}
{{Harmonics in equal|60|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 60edo (continued)}}


; 255ed19
; [[ed12|215ed12]]
* Step size: 20.009{{c}}, octave size: 1200.55{{c}}
Stretching the octave of 215ed12 by around half a cent results in improved primes 3, 5 and 7, but worse primes 2, 11 and 13. This approximates all harmonics up to 16 within 9.44{{c}}. The tuning 215ed12 does this.
{{Harmonics in equal|215|12|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 215ed12}}
{{Harmonics in equal|215|12|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 215ed12 (continued)}}


The tuning [[255ed19]] is 60edo but with slightly ''compressed'' octaves.  
; [[WE|60et, 13-limit WE tuning]] / [[155ed6]]
* Step size: 20.013{{c}}, octave size: 1200.78{{c}}
Stretching the octave of 60edo by just under a cent results in improved primes 3, 5, 7 and 11, but worse primes 2 and 13. This approximates all harmonics up to 16 within 8.63{{c}}. Its 13-limit WE tuning and 13-limit [[TE]] tuning both do this. So does 155ed6 whose octaves differ by only 0.02{{c}}.
{{Harmonics in cet|20.013|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 60et, 13-limit WE tuning}}
{{Harmonics in cet|20.013|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 60et, 13-limit WE tuning (continued)}}


It induces a relatively large improvement to 11/1, at the cost of worsening of every smaller prime. It also causes the patent val for 7/1 to flip from 168 steps to 169.
; [[95edt]]
{{Harmonics in equal|255|19|1|intervals=prime|columns=11|collapsed=1}}
* Step size: 20.021{{c}}, octave size: 1201.23{{c}}
Stretching the octave of 60edo by just over a cent results in improved primes 3, 5, 7 and 11, but worse primes 2 and 13. This approximates all harmonics up to 16 within 7.06{{c}}. The tuning 95edt does this.
{{Harmonics in equal|95|3|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 95edt}}
{{Harmonics in equal|95|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 95edt (continued)}}


; 208ed11
; [[zpi|301zpi]]
* Step size: 20.027{{c}}, octave size: 1201.62{{c}}
Stretching the octave of 60edo by around 1.5{{c}} results in improved primes 3, 5, 7, 11 and 13, but worse primes 2. This approximates all harmonics up to 16 within 6.48{{c}}. The tuning 301zpi does this.
{{Harmonics in cet|20.027|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 301zpi}}
{{Harmonics in cet| 20.027 |intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 301zpi (continued)}}


The tuning [[208ed11]] is 60edo but with slightly ''compressed'' octaves.  
; [[139ed5]]  
* Step size: 20.045{{c}}, octave size: 1202.73{{c}}
Stretching the octave of 60edo by a little under{{c}} results in improved primes 5, 7 and 11, but worse primes 2, 3 and 13. This approximates all harmonics up to 16 within 9.56{{c}}. The tuning 139ed5 does this.
{{Harmonics in equal|139|5|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 139ed5}}
{{Harmonics in equal|139|5|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 139ed5 (continued)}}


It induces a relatively large improvement to 7/1 and 11/1, at the cost of worsening of 2/1, 3/1 and 5/1. It also causes the patent val to flip for 5/1, 7/1 and 17/1.
; [[35edf]]
{{Harmonics in equal|208|11|1|intervals=prime|columns=11|collapsed=1}}
* Step size: 20.056{{c}}, octave size: 1203.35{{c}}
Stretching the octave of 60edo by a little over 3{{c}} results in improved primes 5, 7 and 11 but worse primes 2, 3 and 13. This approximates all harmonics up to 16 within 10.00{{c}}. The tuning 35edf does this.
{{Harmonics in equal|35|3|2|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 35edf}}
{{Harmonics in equal|35|3|2|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 35edf (continued)}}


; 272ed23
== Scales ==
 
* [[5- to 10-tone scales in 60edo]]
The tuning [[272ed23]] is 60edo but with slightly ''compressed'' octaves.
* Amulet{{idiosyncratic}} (approximated from [[25edo]], subset of [[magic]]): 5 2 5 5 2 5 7 5 5 2 5 7 5
 
* Approximations of [[gamelan]] scales:
It induces a relatively large improvement to 7/1 and 11/1, at the cost of worsening of 2/1, 3/1 and 5/1. It also causes the patent val to flip for 5/1, 7/1, 13/1 and 17/1.
** 5-tone pelog: 6 8 20 5 21
 
** 7-tone pelog: 6 8 12 8 5 14 7
These characteristics, particularly the flipped 5/1 and 13/1, are preferred over pure-octaves 60edo for [[catnip]] temperament specifically. They change catnip's [[wart]]s from 60cf to i272dg (later letters in the alphabet are better). Given this, 272ed23 can be thought of as a 60edo clone tailor-made for catnip.
** 5-tone slendro: 12 12 12 12 12
{{Harmonics in equal|272|23|1|intervals=prime|columns=11|collapsed=1}}


== Instruments ==
== Instruments ==
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